Differential and Integral Equations

Optimal control problems in spaces of functions of bounded variation

J. P. Raymond

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We study optimal control problems governed by differential systems involving measures as controls and where the state vector is a function of bounded variation. Such problems are defined as extensions to $BV$-spaces of classical problems which do not admit solutions in the class of absolutely continuous functions. We give a definition of generalized solutions for differential systems with measures, for which we prove a stability result for the weak-star topology of measures. We next prove existence of $BV$-solutions for control problems. A relaxation theorem is given: the classical control problem defined for $AC$-functions and the control problem extended to $BV$-spaces have the same value and the solutions of extended problem are cluster points of minimizing sequences for the initial problem. We finally characterize $BV$-solutions of control problems by means of Lipschitz solutions of an auxiliary control problem.

Article information

Differential Integral Equations, Volume 10, Number 1 (1997), 105-136.

First available in Project Euclid: 6 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49J15: Optimal control problems involving ordinary differential equations
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation


Raymond, J. P. Optimal control problems in spaces of functions of bounded variation. Differential Integral Equations 10 (1997), no. 1, 105--136. https://projecteuclid.org/euclid.die/1367846886

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